In As They Say in French: Other Times, Other Customs, one of the essays published in My Ten Years in a Quandary, and How They Grew (New York: Harper and Brothers, 1936), the U.S. humorist Robert Charles Benchley (1889-1945) wrote:

Speaking of affecting British mannerisms and habits, who remembers when cuffs on a man’s trousers brought down the jibe: “It’s raining in London?”

The earliest occurrences of the obsolete American-English phrase it’s raining in London that I have found date from 1885 and indicate that it originated in New York City, or in stereotypes of the upper-class New Yorker.

The earliest occurrence is from the column All Sorts, in The Times-Democrat (New Orleans, Louisiana) of Friday 20^th February 1885:

So little is there of interest in Wall street that even a poor exchange of compliments between two dudes who have recently appeared in speculative circles makes people laugh. It was a bright, clear, frosty morning when they met. “Ah,” said one, “your trousers are tucked up.” “I know it, deah boy,” was the reply. “Oh, it’s raining in London, then, old chappie; I didn’t know,” lisped the other.—New York Sun.

The second-earliest occurrence that I have found is from The Trenton Times (Trenton, New Jersey) of Sunday 5^th April 1885:

Raining in London.
[Chicago Herald “Walks and Talks.”]

“I heard a story in New York,” said Emory Storrs ¹, “the other day that rather amused me. You know the Knickerbocker club ² there is the nursery of the Anglo-maniac. You don’t find anything at the Knickerbocker club but b. and s. ³, as they call it, and English literature. English newspapers are daily filed; footmen and tigers abound, and the whole atmosphere of the place is the one-eye glass order. A friend of mine, Capt. Bacon, was coming up the street in front of the club-house, when a young man whom he knew came out. “Why, Jack,” he said, “what’s the matter? Got your trousers rolled up and an umbrella spread. Why, it is a bright day—what does it mean?”
“Yes, dear boy. Sun here, you know, but they’ve just got a cable in the club-house that it’s rainin’ in Lunnon. You see?”

¹ This probably refers to Emory A. Storrs (1834-1885), a Chicago lawyer, orator and Republican politician.
² Founded in 1871, the Knickerbocker Club is a gentlemen’s club in New York City.
³ b. and s.: brandy and soda.

In the following from the Salt Lake Evening Democrat (Salt Lake City, Utah) of Monday 20^th April 1885, the very same anecdote was reportedly told by C. R. Barratt, a local dealer in furniture, on his return from a trip to the east of the USA:

“The proper thing now is to ape our next of kin across the water. The Knickerbocker Club of New York has the complaint worse than all others, and has attained nearest to perfection in the matter. A few days since, a gentleman leaving the Club saw a young acquaintance step on to the sidewalk with umbrella raised and pants rolled up. ‘What does this mean?’ he said; ‘the day is bright and the streets dry, and you have your umbrella spread and your pants turned up?’ ‘Yes, my deah boy, but they have just had a cable at the Club from London, you know, and its [sic] raining in London, you know, deah boy;’ replied the devotee of the Club.”

The same anecdote reappears in the following from The Courier-Journal (Louisville, Kentucky) of Sunday 10^th May 1885—this article was reprinted in many U.S. newspapers in June 1885:

AMERICAN “ARISTOCRACY.”
Anglomania and the Rage For Foreign Titles.
“Family Tree”—Progress of Snobbery.
Its Ignominious Maturity at a Foreign Court.
[Correspondence of the Courier-Journal.]

New York, May 7.—In this country we have no aristocracy, according to the European acceptation of the word, and what is more, we do not desire to create one. The rewards which genius and labor bring to the sons and daughters of this republic are sufficient for all who stand up for true Americanism.
It is amusing to watch the efforts which are made by some of our young swells to imitate the manners and eccentricities of English people who move in high society. Anglomania, as this mild form of idiocy is called, has of late years taken possession of a large portion of our dude population, and if this class of brainless, faultlessly-attired youths could have their way an American aristocracy would quickly become a social entity.
A few days ago your representative encountered a would-be American nobleman, walking in the street, with his trousers tucked up and an open umbrella over his head, although the flagged sideway was scrupulously clean, and there was not a cloud visible in the sky.
“Hello,” said the scribe, addressing the Anglo-American dude, “what’s up now, Charley? The sun is not so hot that you need carry an open parasol.”
“Aw, my dear boy,” languidly replied the swell, “you really can not understand it, but ’tis beastly weather.”
“Why, on the contrary, ’tis delightful. A finer day I have not experienced this year.”
“Aw, my dear boy, you think so, but I assure you ’tis raining in a beastly manner.”
“Raining! Where, might I ask you? Look up at the sky and undeceive yourself.”
“Aw, I know all that sort of thing, but fact is, we’ve had a cablegram at the club a few moments ago saying it was raining in London, you know, and, fact is, we fellows who are about to leave for Europe in a few days have already completely identified ourselves with the mother country. Ta, ta.”

The phrase in turn gave rise to raining in London style, describing trousers “long enough to turn up”—as in this advertisement for Smith, Gray & Co., published in The Brooklyn Daily Eagle (Brooklyn, New York) of Monday 26^th June 1893:

A Very Proper Suit

Blue serge coat. White duck trousers, long enough to turn up (raining in London style). Negligee shirt. One of those thin scarfs. Leather belt. Broad brim SENNIT or light Alpine hat. It costs very little money to occupy this suit.

The phrase raining in London style also occurs in the following, published in many U.S. newspapers from November 1910 to January 1911—for example in The Lexington Gazette (Lexington, Virginia) of Wednesday 4^th January:

Will They Wear This?

New York tailors put their heads together recently and evolved the suit pictured, which they have named the suffragette costume. But it remains to be seen whether there is a suffragette sufficiently advanced to appear in this reproduction of father’s togs.
The suit is of gray mannish suiting and there are pockets enough to delight the heart of femininity. Just think of it, six of them, and the trousers are turned up in the approved “raining in London” style. What more could the suffragette want in the way of an equal suffrage costume?

Photo by American Press Association.
THE SUFFRAGETTE SUIT.